All Hebbian rules which are discussed in the
literature of neuromorphic nets are embedded within the principle
of ultrametricity corresponding directly to hierarchical
structures. In order to describe and to model simultaneously
distributed parallel neural activities as they occur in
heterarchically organized systems (self-referentiality) which
cannot be linearized, a formal system for an adequate description
of structural circularities and ambiguities is necessary. A basis
for such a formal system is given by the theory of poly-
contexturality, in which multi-negational operators regulate the
duality principles of complementarity, and transjunctional
operators produce muiti-simultaneous heterarchical structures.

Introduction

One of the criticisms most often heard about
Artifcial Intelligence (AI) is that machines cannot be called
intelligent until they are able to learn to do new things and to
adapt to new situations, rather than simply doing as they were
told to do. There can be little question that adaptibility to new
surroundings and to solve new problems is an important
characteristics of intelligent entities.

Rather than discussing in advance whether it is
possible for computers to "learn", it is more
enlightening to try to describe what activities we mean when we
say "learning" and what algorithms are necessary to
model those activities. One of the very general standard
definitions of learning is "a change in performance (for the
better, as a function of experience)", a definition which is
rather black-box-like and uninformative for the design of any
technical device being able to learn.

In the following discussion we will restrict ourselves to a cybernetical
description of learning processes as it was discussed, for
example, by Bateson (Bateson, 1972) using
Russell´s theory of (logical) types as descriptive basis for a
classification of different learning processes. Despite of its
usefulness and success in various scientific disciplines,
Bateson´s model is strongly hampered by the use of the theory of
logical types especially if the design of technical entities is
envisaged. The shortcomings of this learning model have been
discussed elsewhere (von
Goldammer & Kaehr, 1989)and
will not be repeated here.

The significance of Bateson´s studies is
founded methodologically by their logical analysis and
description of communication processes such as ´learning´.
Keeping the phenomenological part of his classification, we will
distinguish between three categories of ´learning´:

Zero learning

Phenomena which approach this degree of
simplicity occur in various contexts. A technical example is
given by electronic circuits, where the circuit structure itself
is not subject to changes resulting from the passage of
electrical signals within the circuit, i.e., where the causal
links between ´stimulus´ and ´response´ are so to speak
´soldered in´. An intellectually interesting example for ´zero
learning´ is the mathematical fiction of a "player" in
a Von Neumannian game. Without going into details the essential
point of ´zero learning´ is that for this "player"
the principle of "trial and error" is excluded, i.e ,
´trial and error´ learning does not belong to the repertory of
the Von Neumannian "player". Although the meaning of
the word ´error´ is not trivial and will be examined below,
there is a sense in which the "player" can be wrong.
For example, he may base a decision upon probabilistic
consideration and then make a move which, in the light of the
limited available information, was probably right. When more
information becomes available, he may discover that the move was
wrong. But this discovery contributes nothing to his future
skill, i.e., the discovery that he was wrong in the particular
instance has no bearing upon future instances. When the same
problem returns at a later time, he will correctly go through the
same computations and reach the same decisions. An algorithm for
´zero learning´ in the sense of Bateson´s classification is
the ´delta rule´ or ´Widrow-Hoff-rule´ (with a teacher),
where the amount of learning is proportional to the difference
between the actual activation achieved and the target activation
provided by a teacher. In other words, if the entity gives at
Time_2 a different response from what it gave at Time_1, the
changes (adaption of the system) in this case are caused by a
teacher and not by the system itself.

Learning_I or 1st
order learning

While ´zero learning´ by definition is
charactenzed by specifity of response, which - right or wrong -
is not subject to corrections, ´learning_I´ in this terminology
is the change in specifity of response by correction of errors of
choice within a set of alternatives.

All ´Hebbian
rules´ which are discussed in the
literature on neuromorphic nets such as the ´Hopfield´-,
´Boltzmann´-, ´Cauchy´-model and others, which originate from
physics, belong to the category of ´1storder learning´. These rules are related
to changes of the data structure within the learning system. Such
processes correspond to Hebb´s principle of self-organisation,
i.e., the internal organisation or the data-structure (electrical
voltages, currents, etc.) is self-organizing. Thus the
behavioristic principle of ´trial and error´ belongs to the
repertory of ´1st order learning´. In a cybernetical sense neuromorphic
nets are classical I/O-systems with implemented feedback
algorithms which are organized ultrametrically (Rammal, Toulouse &
Virasoro, 1986).One of the
defining relations of metricity is the triangle inequality,

(1)

The notion of ultrametricity is based on the
stronger inequality

(2)

A general
connection between indexed hierarchies
and ultrametrics, which is clearly visible on the classification
tree, was rigorously proved by Benzécri(1984).From
a logical point of view, ultrametricity directly corresponds to
the transitivity relation.

So far all
neuromorphic models discussed in the
literature lead to ultrametricity, the simplest possible
non-trivial organisation of states (Parisi, 1987), and therefore all those nets represent
hierarchically structured models. However, there is one
exception: the neural nets described ´topo-logically´ by
McCulloch in 'A Heterarchy
of Values Determined by the Topology of Nervous Nets'(McCulloch,
1945).A detailed logical
interpretation of this study has been given elsewhere (Kaehr
& von Goldammer, 1988) and
will be discussed very shortly in the next sections.

Apart
from McCulloch´s nets all other neural
models and learning algorithms belong to the field of classical 1st order cybernetics
where all the methods of mathematics based on classical logic are
sufficient for a formal description and the design of algorithms
capable for pattern recognition from a noisy background
(´order-from-disorder´). For the field of robotics this
situation still corresponds to the case of a ´structured
environment´ which is part of the robot controlling program and
consequently does not represent an environment from the viewpoint
of a robot, i.e., a ´structured environment´ is the label for
noisy objects and relations (von
Goldammer & Kaehr, 1989).

While learning_I in the present classification
is a process characterized by correction of errors within a set
of alternatives, learning_II is defined (in this terminology) as
label for all changes in the process of learning_I. For phenomena
of this order various terms have been proposed in the literature
of learning theories, such as ´learning to learn´ or ´set
learning´ (Harlow,
1949).In the terminology of
Bateson, it is the corrective change of the sets of alternatives
which distinguishes ´2nd order learning´ form ´1st order learning´ processes where the corrective change
occurs within a set of alternatives. More technically spoken.
this means that not only the (internal) data structure but also
the algorithm, which defines the structure of the system, changes
simultaneously during such an autonomous learning process. I.e.,
for ´1st
order learning´ it is the variation of the internal organisation
of the data structure which is self-organizing, whereas it is the
relationship between the system (e.g., a robot) and its
environment which is of self-organizing nature for ´2nd order
learning´ processes. This relationship represents a basic
requirement for any description of technical or living systems
acting, for example, in an ´unstructured environment´. Hence it
follows the necessity for any technical design of
self-referentiality to model the process of distinction between a
system and its environment Therefore ´2nd order learning´ differs basically from ´1st order (adaptive)
learning´ as simulated by Hebbian algorithms with their
´causally connected´ way of linkage between domain and internal
structure. In a self-referential process an image of the system
and its environment is produced by the system itself. viz.,

(3a)

(3b)

It is this twofold distinction and hence heterarchical conceptuality which
leads to fundamental difficulties if a formal representation for
an adequate construction plan of corresponding technical devices
is envisaged within the framework of classical logic. Only if the
sytem´s representation is restricted to aspects of itself (self-
representation of reflective architectures) no logical problems
will be produced (cf. ´problems of bootstrapping´; Maes, 1987).

´Learning´ in an ´unstructured
environment´ (2nd order learning) comprises at least two simultaneously
interacting processes:

Both processes are complementary to each other,
i.e., neither of the two can be considered or described
separately. Thus the operator (program) of the volitive process
becomes the operand (data structure) of the cognitive system and
what has been operator of the cognitive process may change into
an operand of the volitive system. Such simultaneously interacting processes
constitute a higher order of circularity (´chiasmus´) and
parallelism which neither can be reduced to linearity (sequemtial
processes) nor can be represented within the linguistic framework
of any classical logical system without producing antinomies
(circularities). However, computational reflection belongs to the
cognitive aspect of behavior whereas volitive aspects usually are
neglected.

Figure 1: Logical
representation of ´self-referentiality´.

( a )circularity
in a classical mono-logical system;

(
b ) distributed circularity on two
contextures;

(
c ) composition of Fig.1b

Figure la illustrates the circularity arising
within a classical logical representation of self-referentiality
as given by relation (3b). This situation results directly from
the inversion of the relationship between operator_&_operand
and operand_&_operator within one logical system. Thus, there
appears no more distinction between operator and operand within
the logical domain which is constitutive for any dichotomic
system. Following Russell´s paradox, the antinomical situation
of this graphic metaphor can easily be logified by the following
equations, where circularity is caused by the substitution of O
by O0 during the transition from (4b) to
(4c):

(4a)

(4b)

(4c)

The correlation
between circularlties and antinomies also applies to other
examples.

The independent variable O0, the ´primary
argument´ has disappeared in (5b). O00 expresses an indefinite recursion of the operator O.
Any indefinite recursion within expression (5b) can be replaced
by O00, as
is indicated in (5b), resulting in (5c). If there are values O00,i (i=1,2,... , n) that satisfy eq. (5c) these values are
called ´eigen-values´

Ei
= O00,i

leading again to closure as in Fig.la, when the
operator is represented by an infinite chain of eigen-values.
This is symbolized by eq. (5d). Although this formalism is useful
on a descriptive level, it is completely unsuited for an
engineering design, because of the central part that ´infinity´
plays as constituent in this representation.

Summarizing in
short we are faced with the obvious
need of a formalism which allows the description and the
engineering design of self-referential processes of autonomous
systems, which are characterized by self- organisation between
the system and its environment. resulting in heterarchically
structured organisations. A theoretical basis for such
constructions is provided by the ´theory of
poly-contexturality´ representing a formal and operative system
of mathematical logic which has been developed first by Günther
(Günther, 1980)and was continued in the following by Kaehr (Kaehr, 1981).

Towards cognitive modeling

A ´contexture´ is a logical domain where all
classical logical rules hold rigorously. The essential point of
´poly-contexturality´ results from the mediation by order and
exchange relations between different (at least three)
contextures. Le the logical domains or contextures do not exist
in isolation, but are mediated with each other by new and
non-classical logical operators, such as for example the ´transjunction´,
which allows the modeling of a bifurcation from one logical
domain into at least two parallel, simultaneously existing
contextures.

In contrast to Fig. la two contextures L1,2 are depicted in Fig.1b in such a way that the relation
between the operator and the operand is distributed among two
(indexed) contextures. Fig. 1c represents the composition of the
distributed relations in Fig. 1b. I.e., circularity is
distributed among two logical domains if the meaning of the terms
will be retained during the transitions from one domain to
another. On the other hand, the relationship between the
operators and operands is distributed on two logical domains and
therefore it escapes any circularity, provided the individual
process will be discriminated during transitions between
different contextures. This connection between operator and
operand has been called ´proemial-relationship´ (Günther, 1980).Such an interchange, i.e., the distribution
and mediation of domains is designated as ´heterarchy´
(heteros = the other and archain = the rule). Heterarchically
organized structures or processes belong to the category of
autonomous and not to the class of I/O-systems. In the
terminology of ´poly-contexturality´, heterarchy is constituted
inter-contextural whereas intra-contextural processes are
hierarchically structured, which means that intra-contextural,
i.e., within the logic of one contexture, the transitivity law
holds rigorously, as do all classical logical rules. Thus a
parallelism is constituted by a (heterarchically) distributed
circularity of the operator and operand which is no longer
reducible to linearity (as a process of sequential steps) as it
is always possible for the purely hierarchically organized models
of neuromorphic nets based on Hebbian learning algorithms.

Since
Russell's theory of logical types is
exclusively hierarchically structured, no mediation between
different equally ranked types exists. Therefore any modeling of
simultaneity in the sense as discussed above is ruled out in
principle which strongly limitates the technical application of
Bateson´s analysis (von
Goldammer & Kaehr, 1989).

Poly-contexturality

In the introduction it has already been
mentioned that the meaning of the word ´error´ is not
trivial especially if it will be used for a description of
learning processes. This problem becomes evident for example from
the pattern in Fig.2, where the exchange relation causes
ambiguity between the logical terms ´true´ and ´false´.I.e. a
proposition may be ´true´ in one contexture and ´false´ in
another one depending on the respective point of view (context or
more precisely contexture).

The scheme corresponds to a three-contextural
logical system, the lowest meaningful contexture in the
poly-contextural theory. While classical logic is defined between
any two values resulting in a contexture, the case with three
values, where the third is not placed between
´true´ and ´false´ but beyond ´true´ and ´false´,
three two-valued logical systems are generated to which three
contextures are assigned.Four values define six logical domains,
and in general with m values (m over 2) two- valued logical
systems are created. The logical Systems or contextures defined
in this way do not coexist in isolation but are mediated with
each other, as is reflected by the scheme of a three-contextural
system in Fig. 2.

Figure 2: Three logical contextures: L(3)=
(L1, L2, L3)

( a ) T, true; F, false;

( b ) short notion of
the mediation between L1, L2, L3

The simultaneity of parallel distributed
processes in poly-contextural systems splits up into two
different types. First there is simultaneity of contextures
without cooperative interactions in between; the contextures,
however, are still mediated with each other. Between the
operators of each contexture, there exist the relationships of
identity, permutation, or reduction. In a three-contextural
system as it is depicted in Fig.2, this is represented in the
following diagrams (7)-(10); secondly, the transjunctional
operator , which allows the modeling of bifurcation from one
logical domain into at least two parallel simultaneously existing
contextures, will be introduced briefly in the diagram (12). For
better understanding the truth table for the logical operations:
conjunction , disjunction , implications , transjunction ,
and two negations (N1,N2) will be presented. In order to simplify the
notation, the following abbreviations have been introduced:

{ T1,
T3 } := T1,3;
{ F1, T2
} := F1,2; { F2,
F3 } := F2,3

(6)

For more details and technical elaboration of the poly-contextural
logic, which cannot be given here, it is referred to the
literature (Kaehr,
1981).

Table 1a will help to elucidate the truth table
not only for the conjunction. but also for the other operators.
Table lb shows the result for the conjunction, while Table 1c
displays a more condensed notation of Table 1b. Table 2 gives the
corresponding version for the disjunction, negation, implication,
and transjunction in L1(see below), which can be derived very
easily with the help of Table la.

( a )

( b )

L1

L2

L3

L1

L2

L3

X1

Y1

X2

Y2

X3

Y3

T1

T1

T3

T3

T1

T3

F1

T1

F1

F3

T3

F3

T1

F1

F1

F1

F1

F2

F2

F1

F2

F2

F2

F2

T3

F3

F3

F2

F2

F2

F2

F2

F3

F3

F2

F3

Table 1 : Truth
table for the conjunction:

(a )
auxilitary table for the construction of (b);

(b ) table of the conjunction

(
c )

T1,3 F1,2
F2,3

T1, 3

F1, 2

F2,3

Table 1c) condensed
version of table 1b)

( a )

( b )

( c )

T1,3

F1,2

F2,3

X

N1X

N2X

T1,3

F1,2

F2,3

T1,3

T1,3

T1

T3

T1,3

F1,2

F3,1

T1,3

T1,3

F1

F3

F1,2

T1

F1,2

F2

F1,2

T1,3

F3,2

F1,2

T1

T1,3

F3

F2,3

T3

F2

F2,3

F2,3

F3,2

F2,1

F2,3

T3

T3

T1,3

Table 2 : Truth
table for (a) the disjunction, (b) the negations, and

(c) the implication.

Permutation

The negational operator N in poly-contextural systems
not only negates its logical domain, but also permutates the
neighboring contextures:

negation in system L1
:

(7)

with :

X

X

T1

F1

F1

T1

negation in system L2
:

(8)

with :

X

X

T2

F2

F2

T2

Identity

Since NZ1 is a superposition of N1 and N2 it produces identity in the 6th step no matter where
one starts; this may be seen from diagram (9).

These relations not only hold for univariate
operators but also for bivariate operators.

cycle of negation NZ1
:

(9)

Reduction

Diagram (10) symbolizes that reduction in L2 occurs caused by R1. In anlogy for R2 the reduction is in L3.

with

For disjunction, see Table
2a.

(10)

Transjunction

Cooperative interactions which are modeled by
transjunctional operations in poly-contextural systems are
defined in a way that an operation in one contexture necessarily
involves other operations in the neighboring contextures
initiated by the transjunction which causes a bifurcation of its
own contexture mapping into the neighbored contextures
independently from the operators working in the corresponding
contextures. The corresponding formula with a transjunction

(11)

is defined in the following diagram (cf. Table
2d):

with

(12)

In this case one has a transjunction in contexture 1 and parallel conjunctions in 2 and 3.
Generally transjunctions together with conjunctions and/or
disjunctions within the neighboring contextures are possible.

Duality principles of complementary in
multi-negational systems

Since the poly-contextural logic is a
multi-negational system, some laws of multi-negation will be
introduced briefly.

two negations and three contextures

negations:

for i = 1, 2

(13)

three negations and six contextures

negations:

for i = 1, 2, 3

(14)

cycles of negations

On the basis of the relations (13), (14) and
the the substitution rule different cycles of negations can be
deduced as for exmple:

or

(15)

Such equivalences are of minor importance for
3-valued systems. For four- and higher-valued systems, however,
they take on significance, since it may become important to know
whether a certain goal of a reflection process may be reached by
different series of negations and which of them are significant.

DeMorgan´s formulae in a
multi-negational system

In classical logical systems duality holds
intra-contextural as indicated by DeMorgan´s formulae for the
disjunction and conjunction:

In transciassical systems there is a
distribution of systems of dualities. Multi-duality in
poly-contextural systems results from mediation of conjunctions,
disjunctions, and negations. For three contextural systems this
may be introduced by the following scheme:

3 contextures:

with negations N1, N2
and disjunctions

with

(16)

The duality operators Di are defined
through the negations Ni, viz.,

or in short:

;

d1 : duality in
L1

(17)

d2 : duality in
L2

Diagram of a 3-duality of
conjunctions and disjunctions:

(18)

For more details and technical elaboration of
the poly-contextural logic, which would go beyond the limits of
the present study, the literature should be consulted (Kaehr,
1981).

The brain which is a self-referential system
par excellence, always interacts with its own states, i.e., it is
a completely closed system (Maturana
& Varela, 1972).It is this
operational closure of the brain functions which implies the
problem of circularity (self-referentiality).

The intention of the present contribution was
to point to the possibility of modeling in detail cognitive
processes without the problem of antinomies. From the viewpoint
of poly-contexturality, however, operational closure only
represents a phenomenon of secondary interest; it is the
distribution, the topology of contextures functions which is of
primary significance.

Thus the complement between the operational
closure and the topologically distributed brain functions -
typical for all heterarchically structured organisations - can be
modeled in an adequate way using the theory of poly-
contexturality, which is characterized by its distribution and
mediation of logical systems.